Optimal. Leaf size=24 \[ \frac {2 \log (x) \left (\left (-1-e^{x/5}+x\right )^4+\log (x)\right )}{x} \]
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Rubi [B] Leaf count is larger than twice the leaf count of optimal. \(166\) vs. \(2(24)=48\).
time = 1.97, antiderivative size = 166, normalized size of antiderivative = 6.92, number of steps
used = 53, number of rules used = 15, integrand size = 198, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.076, Rules used =
{12, 14, 2326, 6874, 2230, 2225, 2208, 2209, 2207, 2634, 45, 2404, 2332, 2341, 2342}
\begin {gather*} 2 x^3 \log (x)-8 e^{x/5} x^2 \log (x)-8 x^2 \log (x)+\frac {8 e^{3 x/5} \left (x \log (x)-x^2 \log (x)\right )}{x^2}+\frac {2 \log ^2(x)}{x}+24 e^{x/5} x \log (x)+12 e^{2 x/5} x \log (x)+12 x \log (x)-24 e^{x/5} \log (x)-24 e^{2 x/5} \log (x)-8 \log (x)+\frac {8 e^{x/5} \log (x)}{x}+\frac {12 e^{2 x/5} \log (x)}{x}+\frac {2 e^{4 x/5} \log (x)}{x}+\frac {2 \log (x)}{x} \end {gather*}
Antiderivative was successfully verified.
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Rule 12
Rule 14
Rule 45
Rule 2207
Rule 2208
Rule 2209
Rule 2225
Rule 2230
Rule 2326
Rule 2332
Rule 2341
Rule 2342
Rule 2404
Rule 2634
Rule 6874
Rubi steps
\begin {gather*} \begin {aligned} \text {integral} &=\frac {1}{5} \int \frac {10+10 e^{4 x/5}+e^{3 x/5} (40-40 x)-40 x+60 x^2-40 x^3+10 x^4+e^{2 x/5} \left (60-120 x+60 x^2\right )+e^{x/5} \left (40-120 x+120 x^2-40 x^3\right )+\left (10+60 x^2-80 x^3+30 x^4+e^{4 x/5} (-10+8 x)+e^{3 x/5} \left (-40+24 x-24 x^2\right )+e^{2 x/5} \left (-60+24 x+12 x^2+24 x^3\right )+e^{x/5} \left (-40+8 x+96 x^2-56 x^3-8 x^4\right )\right ) \log (x)-10 \log ^2(x)}{x^2} \, dx\\ &=\frac {1}{5} \int \left (\frac {2 e^{4 x/5} (5-5 \log (x)+4 x \log (x))}{x^2}-\frac {8 e^{x/5} (-1+x)^2 \left (-5+5 x+5 \log (x)+9 x \log (x)+x^2 \log (x)\right )}{x^2}+\frac {12 e^{2 x/5} (-1+x) \left (-5+5 x+5 \log (x)+3 x \log (x)+2 x^2 \log (x)\right )}{x^2}-\frac {8 e^{3 x/5} \left (-5+5 x+5 \log (x)-3 x \log (x)+3 x^2 \log (x)\right )}{x^2}+\frac {10 \left (1-4 x+6 x^2-4 x^3+x^4+\log (x)+6 x^2 \log (x)-8 x^3 \log (x)+3 x^4 \log (x)-\log ^2(x)\right )}{x^2}\right ) \, dx\\ &=\frac {2}{5} \int \frac {e^{4 x/5} (5-5 \log (x)+4 x \log (x))}{x^2} \, dx-\frac {8}{5} \int \frac {e^{x/5} (-1+x)^2 \left (-5+5 x+5 \log (x)+9 x \log (x)+x^2 \log (x)\right )}{x^2} \, dx-\frac {8}{5} \int \frac {e^{3 x/5} \left (-5+5 x+5 \log (x)-3 x \log (x)+3 x^2 \log (x)\right )}{x^2} \, dx+2 \int \frac {1-4 x+6 x^2-4 x^3+x^4+\log (x)+6 x^2 \log (x)-8 x^3 \log (x)+3 x^4 \log (x)-\log ^2(x)}{x^2} \, dx+\frac {12}{5} \int \frac {e^{2 x/5} (-1+x) \left (-5+5 x+5 \log (x)+3 x \log (x)+2 x^2 \log (x)\right )}{x^2} \, dx\\ &=\frac {2 e^{4 x/5} \log (x)}{x}+\frac {8 e^{3 x/5} \left (x \log (x)-x^2 \log (x)\right )}{x^2}-\frac {8}{5} \int \left (\frac {5 e^{x/5} (-1+x)^3}{x^2}+\frac {e^{x/5} (-1+x)^2 \left (5+9 x+x^2\right ) \log (x)}{x^2}\right ) \, dx+2 \int \left (\frac {(-1+x)^4}{x^2}+\frac {\left (1+6 x^2-8 x^3+3 x^4\right ) \log (x)}{x^2}-\frac {\log ^2(x)}{x^2}\right ) \, dx+\frac {12}{5} \int \left (\frac {5 e^{2 x/5} (-1+x)^2}{x^2}+\frac {e^{2 x/5} (-1+x) \left (5+3 x+2 x^2\right ) \log (x)}{x^2}\right ) \, dx\\ &=\frac {2 e^{4 x/5} \log (x)}{x}+\frac {8 e^{3 x/5} \left (x \log (x)-x^2 \log (x)\right )}{x^2}-\frac {8}{5} \int \frac {e^{x/5} (-1+x)^2 \left (5+9 x+x^2\right ) \log (x)}{x^2} \, dx+2 \int \frac {(-1+x)^4}{x^2} \, dx+2 \int \frac {\left (1+6 x^2-8 x^3+3 x^4\right ) \log (x)}{x^2} \, dx-2 \int \frac {\log ^2(x)}{x^2} \, dx+\frac {12}{5} \int \frac {e^{2 x/5} (-1+x) \left (5+3 x+2 x^2\right ) \log (x)}{x^2} \, dx-8 \int \frac {e^{x/5} (-1+x)^3}{x^2} \, dx+12 \int \frac {e^{2 x/5} (-1+x)^2}{x^2} \, dx\\ &=-24 e^{x/5} \log (x)-24 e^{2 x/5} \log (x)+\frac {8 e^{x/5} \log (x)}{x}+\frac {12 e^{2 x/5} \log (x)}{x}+\frac {2 e^{4 x/5} \log (x)}{x}+24 e^{x/5} x \log (x)+12 e^{2 x/5} x \log (x)-8 e^{x/5} x^2 \log (x)+\frac {2 \log ^2(x)}{x}+\frac {8 e^{3 x/5} \left (x \log (x)-x^2 \log (x)\right )}{x^2}+\frac {8}{5} \int \frac {5 e^{x/5} (-1+x)^3}{x^2} \, dx+2 \int \left (6+\frac {1}{x^2}-\frac {4}{x}-4 x+x^2\right ) \, dx+2 \int \left (6 \log (x)+\frac {\log (x)}{x^2}-8 x \log (x)+3 x^2 \log (x)\right ) \, dx-\frac {12}{5} \int \frac {5 e^{2 x/5} (1-x)^2}{x^2} \, dx-4 \int \frac {\log (x)}{x^2} \, dx-8 \int \left (-3 e^{x/5}-\frac {e^{x/5}}{x^2}+\frac {3 e^{x/5}}{x}+e^{x/5} x\right ) \, dx+12 \int \left (e^{2 x/5}+\frac {e^{2 x/5}}{x^2}-\frac {2 e^{2 x/5}}{x}\right ) \, dx\\ &=\frac {2}{x}+12 x-4 x^2+\frac {2 x^3}{3}-8 \log (x)-24 e^{x/5} \log (x)-24 e^{2 x/5} \log (x)+\frac {4 \log (x)}{x}+\frac {8 e^{x/5} \log (x)}{x}+\frac {12 e^{2 x/5} \log (x)}{x}+\frac {2 e^{4 x/5} \log (x)}{x}+24 e^{x/5} x \log (x)+12 e^{2 x/5} x \log (x)-8 e^{x/5} x^2 \log (x)+\frac {2 \log ^2(x)}{x}+\frac {8 e^{3 x/5} \left (x \log (x)-x^2 \log (x)\right )}{x^2}+2 \int \frac {\log (x)}{x^2} \, dx+6 \int x^2 \log (x) \, dx+8 \int \frac {e^{x/5}}{x^2} \, dx+8 \int \frac {e^{x/5} (-1+x)^3}{x^2} \, dx-8 \int e^{x/5} x \, dx+12 \int e^{2 x/5} \, dx+12 \int \frac {e^{2 x/5}}{x^2} \, dx-12 \int \frac {e^{2 x/5} (1-x)^2}{x^2} \, dx+12 \int \log (x) \, dx-16 \int x \log (x) \, dx+24 \int e^{x/5} \, dx-24 \int \frac {e^{x/5}}{x} \, dx-24 \int \frac {e^{2 x/5}}{x} \, dx\\ &=120 e^{x/5}+30 e^{2 x/5}-\frac {8 e^{x/5}}{x}-\frac {12 e^{2 x/5}}{x}-40 e^{x/5} x-24 \text {Ei}\left (\frac {x}{5}\right )-24 \text {Ei}\left (\frac {2 x}{5}\right )-8 \log (x)-24 e^{x/5} \log (x)-24 e^{2 x/5} \log (x)+\frac {2 \log (x)}{x}+\frac {8 e^{x/5} \log (x)}{x}+\frac {12 e^{2 x/5} \log (x)}{x}+\frac {2 e^{4 x/5} \log (x)}{x}+12 x \log (x)+24 e^{x/5} x \log (x)+12 e^{2 x/5} x \log (x)-8 x^2 \log (x)-8 e^{x/5} x^2 \log (x)+2 x^3 \log (x)+\frac {2 \log ^2(x)}{x}+\frac {8 e^{3 x/5} \left (x \log (x)-x^2 \log (x)\right )}{x^2}+\frac {8}{5} \int \frac {e^{x/5}}{x} \, dx+\frac {24}{5} \int \frac {e^{2 x/5}}{x} \, dx+8 \int \left (-3 e^{x/5}-\frac {e^{x/5}}{x^2}+\frac {3 e^{x/5}}{x}+e^{x/5} x\right ) \, dx-12 \int \left (e^{2 x/5}+\frac {e^{2 x/5}}{x^2}-\frac {2 e^{2 x/5}}{x}\right ) \, dx+40 \int e^{x/5} \, dx\\ &=320 e^{x/5}+30 e^{2 x/5}-\frac {8 e^{x/5}}{x}-\frac {12 e^{2 x/5}}{x}-40 e^{x/5} x-\frac {112 \text {Ei}\left (\frac {x}{5}\right )}{5}-\frac {96 \text {Ei}\left (\frac {2 x}{5}\right )}{5}-8 \log (x)-24 e^{x/5} \log (x)-24 e^{2 x/5} \log (x)+\frac {2 \log (x)}{x}+\frac {8 e^{x/5} \log (x)}{x}+\frac {12 e^{2 x/5} \log (x)}{x}+\frac {2 e^{4 x/5} \log (x)}{x}+12 x \log (x)+24 e^{x/5} x \log (x)+12 e^{2 x/5} x \log (x)-8 x^2 \log (x)-8 e^{x/5} x^2 \log (x)+2 x^3 \log (x)+\frac {2 \log ^2(x)}{x}+\frac {8 e^{3 x/5} \left (x \log (x)-x^2 \log (x)\right )}{x^2}-8 \int \frac {e^{x/5}}{x^2} \, dx+8 \int e^{x/5} x \, dx-12 \int e^{2 x/5} \, dx-12 \int \frac {e^{2 x/5}}{x^2} \, dx-24 \int e^{x/5} \, dx+24 \int \frac {e^{x/5}}{x} \, dx+24 \int \frac {e^{2 x/5}}{x} \, dx\\ &=200 e^{x/5}+\frac {8 \text {Ei}\left (\frac {x}{5}\right )}{5}+\frac {24 \text {Ei}\left (\frac {2 x}{5}\right )}{5}-8 \log (x)-24 e^{x/5} \log (x)-24 e^{2 x/5} \log (x)+\frac {2 \log (x)}{x}+\frac {8 e^{x/5} \log (x)}{x}+\frac {12 e^{2 x/5} \log (x)}{x}+\frac {2 e^{4 x/5} \log (x)}{x}+12 x \log (x)+24 e^{x/5} x \log (x)+12 e^{2 x/5} x \log (x)-8 x^2 \log (x)-8 e^{x/5} x^2 \log (x)+2 x^3 \log (x)+\frac {2 \log ^2(x)}{x}+\frac {8 e^{3 x/5} \left (x \log (x)-x^2 \log (x)\right )}{x^2}-\frac {8}{5} \int \frac {e^{x/5}}{x} \, dx-\frac {24}{5} \int \frac {e^{2 x/5}}{x} \, dx-40 \int e^{x/5} \, dx\\ &=-8 \log (x)-24 e^{x/5} \log (x)-24 e^{2 x/5} \log (x)+\frac {2 \log (x)}{x}+\frac {8 e^{x/5} \log (x)}{x}+\frac {12 e^{2 x/5} \log (x)}{x}+\frac {2 e^{4 x/5} \log (x)}{x}+12 x \log (x)+24 e^{x/5} x \log (x)+12 e^{2 x/5} x \log (x)-8 x^2 \log (x)-8 e^{x/5} x^2 \log (x)+2 x^3 \log (x)+\frac {2 \log ^2(x)}{x}+\frac {8 e^{3 x/5} \left (x \log (x)-x^2 \log (x)\right )}{x^2}\\ \end {aligned} \end {gather*}
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Mathematica [A]
time = 0.12, size = 24, normalized size = 1.00 \begin {gather*} \frac {2 \log (x) \left (\left (1+e^{x/5}-x\right )^4+\log (x)\right )}{x} \end {gather*}
Antiderivative was successfully verified.
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Maple [B] Leaf count of result is larger than twice the leaf count of optimal. \(153\) vs.
\(2(21)=42\).
time = 0.23, size = 154, normalized size = 6.42
method | result | size |
risch | \(\frac {2 \ln \left (x \right )^{2}}{x}+\frac {2 \left (x^{4}-4 \,{\mathrm e}^{\frac {x}{5}} x^{3}+6 \,{\mathrm e}^{\frac {2 x}{5}} x^{2}-4 \,{\mathrm e}^{\frac {3 x}{5}} x +{\mathrm e}^{\frac {4 x}{5}}-4 x^{3}+12 \,{\mathrm e}^{\frac {x}{5}} x^{2}-12 \,{\mathrm e}^{\frac {2 x}{5}} x +4 \,{\mathrm e}^{\frac {3 x}{5}}+6 x^{2}-12 x \,{\mathrm e}^{\frac {x}{5}}+6 \,{\mathrm e}^{\frac {2 x}{5}}+4 \,{\mathrm e}^{\frac {x}{5}}+1\right ) \ln \left (x \right )}{x}-8 \ln \left (x \right )\) | \(107\) |
default | \(\frac {2 \ln \left (x \right ) {\mathrm e}^{\frac {4 x}{5}}}{x}+\frac {40 \ln \left (x \right ) {\mathrm e}^{\frac {3 x}{5}}-40 \ln \left (x \right ) {\mathrm e}^{\frac {3 x}{5}} x}{5 x}+\frac {60 \ln \left (x \right ) {\mathrm e}^{\frac {2 x}{5}}-120 \ln \left (x \right ) {\mathrm e}^{\frac {2 x}{5}} x +60 \ln \left (x \right ) {\mathrm e}^{\frac {2 x}{5}} x^{2}}{5 x}+\frac {40 \ln \left (x \right ) {\mathrm e}^{\frac {x}{5}}-120 \ln \left (x \right ) {\mathrm e}^{\frac {x}{5}} x +120 \ln \left (x \right ) x^{2} {\mathrm e}^{\frac {x}{5}}-40 \ln \left (x \right ) {\mathrm e}^{\frac {x}{5}} x^{3}}{5 x}-8 \ln \left (x \right )+\frac {2 \ln \left (x \right )^{2}}{x}+\frac {2 \ln \left (x \right )}{x}+2 x^{3} \ln \left (x \right )-8 x^{2} \ln \left (x \right )+12 x \ln \left (x \right )\) | \(154\) |
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [F]
time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \text {Failed to integrate} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [B] Leaf count of result is larger than twice the leaf count of optimal. 77 vs.
\(2 (21) = 42\).
time = 0.38, size = 77, normalized size = 3.21 \begin {gather*} \frac {2 \, {\left ({\left (x^{4} - 4 \, x^{3} + 6 \, x^{2} - 4 \, {\left (x - 1\right )} e^{\left (\frac {3}{5} \, x\right )} + 6 \, {\left (x^{2} - 2 \, x + 1\right )} e^{\left (\frac {2}{5} \, x\right )} - 4 \, {\left (x^{3} - 3 \, x^{2} + 3 \, x - 1\right )} e^{\left (\frac {1}{5} \, x\right )} - 4 \, x + e^{\left (\frac {4}{5} \, x\right )} + 1\right )} \log \left (x\right ) + \log \left (x\right )^{2}\right )}}{x} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Sympy [B] Leaf count of result is larger than twice the leaf count of optimal. 148 vs.
\(2 (20) = 40\).
time = 0.29, size = 148, normalized size = 6.17 \begin {gather*} - 8 \log {\left (x \right )} + \frac {\left (2 x^{4} - 8 x^{3} + 12 x^{2} + 2\right ) \log {\left (x \right )}}{x} + \frac {2 \log {\left (x \right )}^{2}}{x} + \frac {2 x^{3} e^{\frac {4 x}{5}} \log {\left (x \right )} + \left (- 8 x^{4} \log {\left (x \right )} + 8 x^{3} \log {\left (x \right )}\right ) e^{\frac {3 x}{5}} + \left (12 x^{5} \log {\left (x \right )} - 24 x^{4} \log {\left (x \right )} + 12 x^{3} \log {\left (x \right )}\right ) e^{\frac {2 x}{5}} + \left (- 8 x^{6} \log {\left (x \right )} + 24 x^{5} \log {\left (x \right )} - 24 x^{4} \log {\left (x \right )} + 8 x^{3} \log {\left (x \right )}\right ) e^{\frac {x}{5}}}{x^{4}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Giac [B] Leaf count of result is larger than twice the leaf count of optimal. 285 vs.
\(2 (21) = 42\).
time = 0.43, size = 285, normalized size = 11.88 \begin {gather*} \frac {2 \, {\left (x^{4} \log \left (5\right ) - 4 \, x^{3} e^{\left (\frac {1}{5} \, x\right )} \log \left (5\right ) + x^{4} \log \left (\frac {1}{5} \, x\right ) - 4 \, x^{3} e^{\left (\frac {1}{5} \, x\right )} \log \left (\frac {1}{5} \, x\right ) - 4 \, x^{3} \log \left (5\right ) + 6 \, x^{2} e^{\left (\frac {2}{5} \, x\right )} \log \left (5\right ) + 12 \, x^{2} e^{\left (\frac {1}{5} \, x\right )} \log \left (5\right ) - 4 \, x^{3} \log \left (\frac {1}{5} \, x\right ) + 6 \, x^{2} e^{\left (\frac {2}{5} \, x\right )} \log \left (\frac {1}{5} \, x\right ) + 12 \, x^{2} e^{\left (\frac {1}{5} \, x\right )} \log \left (\frac {1}{5} \, x\right ) + 6 \, x^{2} \log \left (5\right ) - 4 \, x e^{\left (\frac {3}{5} \, x\right )} \log \left (5\right ) - 12 \, x e^{\left (\frac {2}{5} \, x\right )} \log \left (5\right ) - 12 \, x e^{\left (\frac {1}{5} \, x\right )} \log \left (5\right ) + 6 \, x^{2} \log \left (\frac {1}{5} \, x\right ) - 4 \, x e^{\left (\frac {3}{5} \, x\right )} \log \left (\frac {1}{5} \, x\right ) - 12 \, x e^{\left (\frac {2}{5} \, x\right )} \log \left (\frac {1}{5} \, x\right ) - 12 \, x e^{\left (\frac {1}{5} \, x\right )} \log \left (\frac {1}{5} \, x\right ) + e^{\left (\frac {4}{5} \, x\right )} \log \left (5\right ) + 4 \, e^{\left (\frac {3}{5} \, x\right )} \log \left (5\right ) + 6 \, e^{\left (\frac {2}{5} \, x\right )} \log \left (5\right ) + 4 \, e^{\left (\frac {1}{5} \, x\right )} \log \left (5\right ) + \log \left (5\right )^{2} - 4 \, x \log \left (\frac {1}{5} \, x\right ) + e^{\left (\frac {4}{5} \, x\right )} \log \left (\frac {1}{5} \, x\right ) + 4 \, e^{\left (\frac {3}{5} \, x\right )} \log \left (\frac {1}{5} \, x\right ) + 6 \, e^{\left (\frac {2}{5} \, x\right )} \log \left (\frac {1}{5} \, x\right ) + 4 \, e^{\left (\frac {1}{5} \, x\right )} \log \left (\frac {1}{5} \, x\right ) + 2 \, \log \left (5\right ) \log \left (\frac {1}{5} \, x\right ) + \log \left (\frac {1}{5} \, x\right )^{2} + \log \left (5\right ) + \log \left (\frac {1}{5} \, x\right )\right )}}{x} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Mupad [F]
time = 0.00, size = -1, normalized size = -0.04 \begin {gather*} \int \frac {2\,{\mathrm {e}}^{\frac {4\,x}{5}}-8\,x+\frac {{\mathrm {e}}^{\frac {2\,x}{5}}\,\left (60\,x^2-120\,x+60\right )}{5}-\frac {{\mathrm {e}}^{x/5}\,\left (40\,x^3-120\,x^2+120\,x-40\right )}{5}-2\,{\ln \left (x\right )}^2-\frac {{\mathrm {e}}^{\frac {3\,x}{5}}\,\left (40\,x-40\right )}{5}+12\,x^2-8\,x^3+2\,x^4+\frac {\ln \left (x\right )\,\left ({\mathrm {e}}^{\frac {2\,x}{5}}\,\left (24\,x^3+12\,x^2+24\,x-60\right )-{\mathrm {e}}^{\frac {3\,x}{5}}\,\left (24\,x^2-24\,x+40\right )-{\mathrm {e}}^{x/5}\,\left (8\,x^4+56\,x^3-96\,x^2-8\,x+40\right )+{\mathrm {e}}^{\frac {4\,x}{5}}\,\left (8\,x-10\right )+60\,x^2-80\,x^3+30\,x^4+10\right )}{5}+2}{x^2} \,d x \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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